Commit 21ef9f74 by Jim Hefferon

adjust wording of implication discussion in logic slides

parent a0b9fe55
\documentclass[11pt]{article}
\usepackage[T1]{fontenc}
\usepackage[default]{gillius}
\setlength{\parindent}{0em}
\setlength{\parskip}{1ex}
\begin{document}\thispagestyle{empty}
\begin{center}
\Large Pledge
\end{center}
I understand that in this class I must work entirely on my own.
I will not
use additional texts, or the Internet,
or discuss the work with other people in the class,
or people who have taken the class in the past,
or anyone at all.
Please sign and date below the line.
\vspace{2ex}
\begin{flushright}
\begin{tabular}{@{}p{2.5in}p{1in}}
\textit{Name} &\textit{Date} \\
\hline
& \\
& \\
& \\
& \\
& \\
\end{tabular}
\end{flushright}
\end{document}
No preview for this file type
......@@ -7,7 +7,7 @@
\PassOptionsToPackage{usenames,dvipsnames}{xcolor}
% \DeclareGraphicsRule{*}{mps}{*}{}
% \usepackage{../ibl}
\usepackage{../ibl}
% \usepackage{../ibl}
\usepackage{present}
% \usepackage{xr}\externaldocument{../ibl} % read refs from .aux file
% \usepackage{catchfilebetweentags}
......@@ -39,38 +39,69 @@
\subject{Inquiry-based learning}
% This is only inserted into the PDF information catalog. Can be left
% out.
%% \usepackage[T1]{fontenc}
%% \usepackage[default]{gillius}
\begin{document}
% \begin{frame}
% \titlepage
% \end{frame}
\section{Learning}
% ============================================================
\begin{frame}{Learning}
In the education business every few years there
is a new fad.
Usually it says just the opposite of the old fad.
Doesn't anyone \emph{know} anything?
\pause
This is
\href{https://www.youtube.com/watch?v=Yi_5xbd5xdE}{Nadia Comaneci},
shocking everyone with a perfect 10 in the 1976 olympics.
\pause
This is
\href{https://www.youtube.com/watch?v=ggq0xtaTqj8}{Simone Biles}
doing the same event in 2016.
\pause
Somebody knows how people
can learn to do things that are very hard.
\end{frame}
\section{Inquiry-Based Learning}
% ============================================================
\begin{frame}{Discussion}
\begin{enumerate}
\item How do you know when you % \textit{really}
\item How do you know when you
understand something?
\pause
\item What do you expect to remember in twenty years?
\pause
\item How do you learn something new?
\pause
\item As part of learning, what is the value
of making mistakes?
\pause
\item How do we create a learning community,
dedicated to inquiry?
% \pause\medskip
% It must include respect for contributions,
% and for the time and thought they take,
% as well as
% encouragement for risk taking, and an understanding of the value of
% productive failure.
%% \pause
%% \item As part of learning, what is the value
%% of making mistakes?
\end{enumerate}
\end{frame}
\begin{frame}{How to create a learning community?}
A learning community is dedicated to inquiry.
It must have high standards,
an understanding of the difference
between what's pretty good, and what is completely right.
\pause
It must also include respect
for the time and thought that all contributions take,
an understanding of the value of
productive struggle,
and an
encouragement for risk taking.
% \vspace*{1ex plus1fill}
% {\scriptsize (Adapted from Dana Ernst.)}
......@@ -92,20 +123,21 @@ give an example.
\pause
Between class meetings you will have three or four of these
statements to think about.
statements.
You will work on your own, without
using additional texts, or the Internet,
or discussing the work with other people in the class,
or people who have taken the class in the past,
or anyone at all.
At the next meeting, class members will propose solutions and
we will work those through, as a group.
At the next meeting, class members will propose solutions,
on the board, and we will work those through as a group.
\end{frame}
%...........................
% \begin{frame}
% \ExecuteMetaData[../gr3.tex]{GaussJordanReduction}
......
No preview for this file type
% see: https://groups.google.com/forum/?fromgroups#!topic/comp.text.tex/s6z9Ult_zds
\makeatletter\let\ifGm@compatii\relax\makeatother
\documentclass[10pt,t]{beamer}
\documentclass[9pt,t]{beamer}
\usefonttheme{professionalfonts}
\usefonttheme{serif}
\PassOptionsToPackage{pdfpagemode=FullScreen}{hyperref}
......@@ -25,6 +25,9 @@
}
\addheadbox{filler}{\ } % create extra space at top of slide
\hypersetup{colorlinks=true,linkcolor=blue}
\setbeamerfont{title}{series=\bfseries,parent=structure}
\setbeamerfont{frametitle}{series=\bfseries,parent=structure}
\setbeamerfont{section title}{size=\LARGE,series=\bfseries} % doesn't seem to do anything
\title[Foundation of proofs] % (optional, use only with long paper titles)
{Foundation of proofs}
......@@ -293,7 +296,7 @@ We can describe the action of these operators using \alert{truth tables}.
\pause
One advantage of this
notation is that it allows formulas of a complexity that would be awkward in
notation is that it allows formulas more complex than you could say in
a natural language.
For instance,
$(P\vee Q)\wedge \neg(P\wedge Q)$ is hard to express in
......@@ -389,8 +392,8 @@ We model `if $P$ then $Q$' this way.
$T$ &$T$ &$T$
\end{tabular}
\end{center}
(We will address some subtle aspects of this definition below.)
Here $P$ is the \alert{antecedent} while $Q$ is the
(We will speak to some subtle aspects of this definition below.)
Here, $P$ is the \alert{antecedent} while $Q$ is the
\alert{consequent}.
\end{frame}
......@@ -402,7 +405,9 @@ Here $P$ is the \alert{antecedent} while $Q$ is the
\begin{frame}
\frametitle{Bi-implication}
Model `$P$ if and only if $Q$' with this.
We take `$P$ if and only if $Q$' to mean the two have the
same values, `a number~$n$ is divisible by $5$' if and only if
`the number~$n$ ends in $0$ or~$5$'.
\begin{center}
\begin{tabular}{cc|c}
$P$ &$Q$ &$P \leftrightarrow Q$ \\ \hline
......@@ -421,7 +426,7 @@ Mathematicians often write `iff'.
\begin{frame}
\frametitle{All binary operators}
This lists all of the binary logical functions.
We can list all of the binary logical functions.
\begin{center} \small
\begin{tabular}{cc|c}
$P$ &$Q$ &$P$ $\alpha_0$ $Q$ \\ \hline
......@@ -525,7 +530,7 @@ $T$ and $F$.
For instance, $P\wedge Q$ and $Q\wedge P$ are equivalent.
\pause
Another example is that $P\rightarrow Q$ and $\neg Q\rightarrow \neg P$
An important example is that $P\rightarrow Q$ and $\neg Q\rightarrow \neg P$
are equivalent.
\begin{center}
\begin{tabular}{cc|c|ccc}
......@@ -588,36 +593,35 @@ $T$ and $F$.
\begin{frame}
\frametitle{Discussion: our definition of `implies'}
\vspace*{-1ex}
For $P\rightarrow Q$ everyone expects when $P$ is true then $Q$ will follow,
so that if $P$ is~$T$ but $Q$
is~$F$ then the statement as a whole is~$F$.
What about the other cases?
\begin{center}
\begin{tabular}{cc|c}
$P$ &$Q$ &$P \rightarrow Q$ \\ \hline
$F$ &$F$ &$T$ \\
$F$ &$T$ &$T$ \\
$F$ &$F$ &\onslide<3->{$T$} \\
$F$ &$T$ &\onslide<2->{$T$} \\
$T$ &$F$ &$F$ \\
$T$ &$T$ &$T$
$T$ &$T$ &\onslide<4->{$T$}
\end{tabular}
\end{center}
In Mathematics we take the statement
`if Babe Ruth was president then $1+2=4$'
to be true
because its antecedent is false.
\pause
Similarly we take
`if Mallory reached the summit of Everest then $1+2=3$'
to be true because its consequent is true.
\pause
Why define it this way?
Standard mathematical practice defines implication so that
statements like this are true for all real numbers:
Standard mathematical practice defines implication so that, for instance,
this statement is true for all real numbers:
\begin{center}
if $x$ is rational then $x^2$ is rational
\end{center}
(because $x=p/q$ gives $x^2=p^2/q^2$).
\pause Then taking $x=\sqrt{2}$ shows that we need
$F\rightarrow T$ to evaluate to $T$.
\pause Taking $x=\sqrt{2}$ shows that we need
$F\rightarrow T$ to evaluate to $T$.
\pause Take $x=\pi$ to see that we need $F\rightarrow F$ to yield $T$.
\pause For $T\rightarrow T$ take $x=1/2$.
\pause
The intuition is that $P\rightarrow Q$ is a
promise that if $P$ holds then $Q$ must hold also.
If $P$ doesn't hold, that is not a counterexample to the promise.
If $Q$ does hold, that is also not a counterexample.
\end{frame}
% http://www.earlham.edu/~peters/courses/log/mat-imp.htm
\begin{frame}\vspace*{-1ex}
......@@ -632,18 +636,22 @@ $F\rightarrow T$ to evaluate to $T$.
\end{tabular}
\end{center}
\begin{itemize}
\item As noted on the prior slide, our definition does not require that
the antecedent~$P$ causes, or is in any way connected to, the consequent~$Q$.
\item
If the antecedent~$P$ is false then the statement as a whole is true,
said to be \alert{vacuously true}.
If the consequent~$Q$ is true then the statement as a whole is true.
\pause
\item
Thus, we take
`if Babe Ruth was president then $1+2=4$'
to be true, vacuously true.
Similarly, we take
`if Mallory reached the summit of Everest then $1+2=3$'
to be true.
\pause
\item Also noted there is:
(1)~if the antecedent~$P$ is false then the statement as a whole is true,
said to be \alert{vacuously true},
and (2)~if the consequent~$Q$ is true then the statement as a whole is true.
\item In particular, our definition does not require that
the antecedent~$P$ causes, or is in any way connected to, the consequent~$Q$.
\pause
\item The intuition behind implication $P\rightarrow Q$ is that it is a
promise that if $P$ holds then $Q$ must hold also.
If $P$ doesn't hold, that is not a counterexample to the promise.
Likewise, if $Q$ does hold, that is no counterexample, either.
% \pause
% \item If the antecedent~$P$ is true then the statement as a whole has the
% same truth value as the consequent.
......@@ -703,8 +711,9 @@ A
\alert{quantifier} delimits for how many values of the
variable the clause must be true, in order for the statement as a whole to
be true.
Besides `for all'
we will also see
we will also use
`there exists', denoted $\exists$.
The statement
$\exists n\in\N \big[\textit{Odd}(n)\rightarrow\textit{Square}(n)\big]$
......
......@@ -3,7 +3,9 @@
\usepackage[T1]{fontenc} % Font encoding: T1
\usepackage{conc}
\usepackage[T1]{fontenc}
\usepackage[default]{gillius}
% \usepackage{conc}
% Make TeX not give "no such font size" errors
% \usepackage{type1cm}
% \usepackage{exscale} % uncommenting this makes \infty disappear
......@@ -15,6 +17,16 @@
% \setmonofont[Scale=MatchLowercase]{Dina ttf 10px}
% % \DeclareGraphicsRule{*}{mps}{}{}
\usepackage{amsmath,amsfonts}
\def\C{\mathbb{C}}
\def\N{\mathbb{N}}
\def\Q{\mathbb{Q}}
\def\R{\mathbb{R}}
\def\Z{\mathbb{Z}}
\newcommand{\divides}{|}
\newcommand{\map}[3]{#1\colon#2\to#3}
% Dash code stolen from tugboat.dtx
\def\thinskip{\hskip 0.16667em\relax}
\def\endash{--}
......@@ -150,7 +162,7 @@
% Some misc definitions
\renewcommand{\qedsymbol}{{\scshape\small QED}}
\renewcommand{\definend}[1]{\textcolor{red}{\textit{#1}}}
\def\definend#1{\textcolor{red}{\textit{#1}}}
\newcommand{\appendrefs}[1]{\relax}
% Make \section{..} show a page all alone.
......@@ -167,5 +179,5 @@
% Vertically center graphics
% ex: \vcenteredhbox{\usegraphics{mygraph.png}}
% From http://tex.stackexchange.com/questions/7219/how-to-vertically-center-two-images
% \newcommand*{\vcenteredhbox}[1]{\begingroup
% \setbox0=\hbox{#1}\parbox{\wd0}{\box0}\endgroup}
\newcommand*{\vcenteredhbox}[1]{\begingroup
\setbox0=\hbox{#1}\parbox{\wd0}{\box0}\endgroup}
Markdown is supported
0% or
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment